1. This would be really hard, and would require a lot of physics to do, and I think there may already be an expensive commercial product for you to compete with. Something that tells you how to adjust your body to spin better. E.g., pitches that tell you something about where to move a hand/arm and/or foot/leg. Should also tell you whether your spin is traveling, and in what direction. You need good GPS reception inside the rink for the last part.
2. An app that shows what pattern you actually skated. Again, needs good GPS reception. This is really the same as some trail GPS programs do. But: add tones to show how you are off from the desired pattern.
3. An app that computes how much money and time you are spending on figure skating. Include session fees, sharpening, lessons, cost-of-driving, other travel expenses, clothing, time spent playing with skatingforums.com, etc. A magic button that makes all the money spent go away.
Oh, wait - skating itself already makes all the money disappear.
4. An app that warns a parent if a child uses their phone (for calls, apps, messages, etc.) instead of skating during an expensive skating session.